Periodicity, chaos and multiple coexisting attractors in a generalized Moore–Spiegel system

Abstract

Introduced in 1966 by Moore and Spiegel, the so called Moore–Spiegel system displays aperiodic dynamics that describes the irregular variability of the luminosity of the stars. The Moore–Spiegel system is defined by a jerk system with a single cubic nonlinearity that is responsible for the chaotic dynamics of the whole system. In this contribution, the dynamics of the generalized Moore-Spiegel system recently investigated by [Letellier and Malasoma, Chaos, Solitons & Fractals 69 (2014) 40–49] is considered. Some fundamental dynamical properties of the model such as fixed points, phase portraits, basins of attraction, bifurcation diagrams, and Lyapunov exponents are investigated. Analysis shows that chaos is achieved via period-doubling and symmetry restoring crisis scenarios. One of the major results of this work is the finding of various windows in the parameters’ space in which two, three, four or six different attractors coexist, depending solely on the choice of initial conditions. This unusual and striking phenomenon has not yet been reported previously in the Moore–Spiegel system, and thus deserves dissemination. Compared to some lower dimensional systems capable of six disconnected coexisting periodic and chaotic attractors reported to date, the Moore–Spiegel system represents one of the simplest and the most ‘elegant’ paradigms. Some PSIM based simulations of the nonlinear dynamics of the system are carried out to validate the results of theoretical analyzes.